zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Homothetic and conformal symmetries of solutions to Einstein’s equations. (English) Zbl 0604.53038
Authors’ abstract: ”We present several results about the nonexistence of solutions of Einstein’s equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spacetimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolve to a spacetime with full isometry.”
Reviewer: I.Gottlieb
MSC:
53B50Applications of local differential geometry to physics
83C20Classes of solutions of equations in general relativity
References:
[1]Gerhardt, C.: H-surfaces in Lorentcian manifolds. Commun. Math. Phys.89, 523 (1983) · Zbl 0519.53056 · doi:10.1007/BF01214742
[2]Marsden, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep.66, 109 (1980) · doi:10.1016/0370-1573(80)90154-4
[3]Löbell, F.: Ber. Verhandl. Sächs Akad. Wiss. Leipzig. Math. Phys. K1.83, 167 (1931)
[4]Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Gravitation: An introduction to current research. Witten, L. (ed.). New York: Wiley 1962
[5]Fisher, A., Marsden, J.: Bull. Am. Math. Soc.79, 995 (1973) · Zbl 0276.43010 · doi:10.1090/S0002-9904-1973-13322-1
[6]Fischer, A., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einstein’s equations. I. One Killing field. Ann. Inst. H. Poincaré33, 147 (1980)
[7]Arms, J., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einstein’s equations. II. Several Killing fields and the Einstein-Yang-Mills equations. Ann. Phys. (N.Y.)144, 81 (1982) · Zbl 0499.53052 · doi:10.1016/0003-4916(82)90105-1
[8]O’Murchadha, N., York, J. W.: Initial-value problem of general relativity. II. Stability of solutions of the initial-value equations. Phys. Rev.D 10, 437 (1974)
[9]Isenberg, J., Nester, J.: Canonical gravity. In: General relativity and gravitation. Vol. 1. Held, A. (ed.). New York, London: Plenum 1980
[10]Fischer, A., Marsden, J.: The initial value problem and the dynamical formulation of general relativity. In: General relativity. Einstein centenary survey. Hawking, S., Israel, W. (eds.). Cambridge: Cambridge University Press 1979
[11]Choquet-Bruhat, Y., York, J.: The Cauchy problem. In: General relativity and gravitation. Vol. 1. Held, A. (ed.). New York, London: Plenum 1980
[12]Berger, B. K.: Homothetic and conformal motions in spacelike slices of solutions of Einstein’s equations. J. Math. Phys.17, 1268 (1976) · doi:10.1063/1.523052
[13]Misner, C., Thorne, K. Wheeler, J.: Gravitation. San Francisco: W. H. Freemann 1973
[14]Hawking, S. W., Ellis, G. F. R.: The large scale structure of spacetime. Cambridge: Cambridge University Press 1983
[15]Eardley, D. M.: Self-similar spacetimes: Geometry and dynamics. Commun. Math. Phys.37, 287 (1974) · doi:10.1007/BF01645943
[16]Yano, K., Bochner, S.: Curvature and Betti numbers. In: Ann. Math. Stud. No.32. Princeton: Princeton University Press 1953
[17]DeWitt, B.: In: Relativity, group and topology. DeWitt, C. DeWitt, B. (eds.) Gordon and Breach 1964
[18]Yano, K.: The theory of Lie derivatives and its applications. New York: Interscience 1957
[19]Garfinkle, D., Tiem, Q. J.: private communication 1985
[20]Schoen, R., Yau, S. -T.: Proof of the positive mass theorem. II. Commun. Math. Phys.79, 231 (1981), and their previous papers cited therein · Zbl 0494.53028 · doi:10.1007/BF01942062
[21]Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381 (1981) · doi:10.1007/BF01208277
[22]Parker, T., Taubes, C. H.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys.84, 223 (1982) · Zbl 0528.58040 · doi:10.1007/BF01208569
[23]Yip, P.: private communication 1985
[24]Maithreyan, T. Eardley, D.: Gravitational collapse of spherically symmetric scale invariant scalar fields. ITP preprint 1985
[25]Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differen. Geom. (To appear).
[26]Isenberg, J., Marsden, J.: Geom. Phys.1, 85 (1984) · Zbl 0589.53071 · doi:10.1016/0393-0440(84)90015-9
[27]Fischer, A., Marsden, J.: Can. J. Math.29, 193 (1977) · Zbl 0358.58006 · doi:10.4153/CJM-1977-019-x
[28]Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations. I. J. Math. Phys.16, 493 (1975) · Zbl 0314.53035 · doi:10.1063/1.522572
[29]Jantzen, R., Rosquist, K.: Adapted Slicings of space-times possessing simply transitive similarity groups. J. Math. Phys.27, 1191 (1986). · Zbl 0595.70002 · doi:10.1063/1.527125
[30]Hsu, L., Wainwright, J.: Self-similar spatially homogeneous cosmologies I: Orthogonal perfect fluid and vacuum solutions. Class. Quantum Gravity. (To appear).
[31]Garfinkle, D.: Asymptotically flat spacetimes have no conformal Killing fields. J. Math. Phys. (To appear).